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Communications in Theoretical Physics

First-Order Transition in a Quantum Ising Model with p -Spin Interactions and a Random Field

Yu‐qiang Ma, Z.D. Wang, Li Sheng, Jian‐Xin Zhu, Chang-De Gong

Published Jun 15, 1997• 0 citations
Publisher

Abstract

We study a quantum transverse Ising model with p -spin interactions in the presence of a random field. The formulation for the free energy of the system is derived by making use of the Suzuki–Trotter approach with the thermodynamic perturbation theory, and the first-order transitions are obtained in the limit p → ∞ .
DOI
10.1088/0253-6102/27/4/419

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We have studied the phase transition behaviour of the random field Ising model in the presence of a transverse (or tunnelling) field. The mean field phase diagram has been studied in detail, and in particular the nature of the transition induced by the tunnelling (transverse) field at zero temperature. A modified hyperscaling relation for the zero-temperature transition has been derived using the Suzuki - Trotter formalism and a modified `Harris criterion'. A mapping of the model to a randomly diluted antiferromagnetic Ising model in uniform longitudinal and transverse field is also given.
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We introduce the transverse Ising model as a prototype for discussing quantum phase transitions. Mean field theory and its application to superconductivity by BCS are discussed, as well as real space renormalization group techniques in one dimension. Next, we introduce the Suzuki-Trotter formalism to show the correspondence between d-dimensional quantum systems and (d + 1)-dimensional classical systems. We then discuss transverse Ising spin glass models, namely the Sherrington-Kirkpatrick (SK) model, the Edwards-Anderson (EA) model and the ±J model. We briefly discuss the mean field, exact diagonalization and quantum Monte Carlo results for their phase diagrams. We discuss the question of replica symmetry restoration in quantum spin glasses due to the possibility of tunneling through the barriers. Finally, we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses.
The theory of dynamical quantum phase transitions represents an attempt to extend the concept of phase transitions to the far from equilibrium regime. While there are many formal analogies to conventional transitions, it is a major question to which extent it is possible to formulate a nonequilibrium counterpart to a Landau-Ginzburg theory. In this paper we take a first step in this direction by constructing an effective free energy for continuous dynamical quantum phase transitions appearing after quantum quenches in the transverse-field Ising chain. Due to unitarity of quantum time evolution this effective free energy becomes a complex quantity transforming the conventional minimization principle of the free energy into a saddle-point equation in the complex plane of the order parameter, which as in equilibrium is the magnetization. We study this effective free energy in the vicinity of the dynamical quantum phase transition by performing an expansion in terms of the complex magnetization and discuss the connections to the equilibrium case. Furthermore, we study the influence of perturbations and signatures of these dynamical quantum phase transitions in spin correlation functions.